Infinity plus one

This depends on what you mean by `infinity'. For example, if you are talking about transfinite ordinals, `infinity' might mean omega (also written omega_0), the first transfinite ordinal. In that case, `infinity plus one' would mean omega + 1, which is well-defined, and is distinct from omega (omega +1 = omega union { omega }). It has the same cardinality, yes, but ordinals are not cardinals.

If the children know their math they could move on to the next step and say, "Yeah? Well, I have 2 to the power of infinity!" 2^infinity has a higher cardinality than simply infinity (usually defined as all real numbers or rational numbers). At this point things get ridiculous (if they haven't already) because there will always be a larger set (the power set), even if you took 2 to the power of infinity an infinite number of times there would still be a larger set. The diagonal argument shows how this is true.

Many a student who has used the basics of algebra, and has solved x + 6 = 2x + 1 and other simple unknowns1 has encountered the concept of infinity, and assumed that the symbol ∞ can be manipulated just like x; and that since infinity cannot be enlarged upon by addition, Infinity plus one equals infinity. Then they tend to use that basis to make fake proofs of nonsense:

In general for any real numbers n and m,
Step 1: ∞ + n = ∞ + m
so
step 2: m = n
Thus any number is equal to any other.

So is arithmetic broken?
Well, no. One is still not equal to zero. Adding up gives consistent results. The problem is that is the first step we assume that infinity does not have a fixed value – it is equal to itself plus one. In the second step we assume that it does have a fixed value – that it can be subtracted from both sides, leaving the equality relation intact. These assumptions are contradictory, and the results obtained may not be correct.

So which step is incorrect? Well, both:
1) It's widely believed that infinity plus one equals infinity, but it's an oversimplification. There is no real number that is equal to itself plus one - zero is the identity element for addition of real numbers, not one. Infinity is not a real number, so addition is not defined the same way upon it.
2) Likewise, subtraction is not defined upon it. The first step is a statement about infinity which has rhetorical force if not mathematical rigour. The second step is a false statement two real numbers.